Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities

and Applied Analysis 3 In this paper, we also consider themultiplicity of solutions of (7). For each l ∈ Z, let l ∗ u = (u n+lT ) n∈Z , ∀u = (u n ) n∈Z , (21) which defines a Z-action on E. By the periodicity of the coefficients, we know that both J and J󸀠 are Z-invariants. Therefore, ifu ∈ E is a critical point of J, so is l∗u. Two critical points u 1 , u 2 ∈ E of J are said to be geometrically distinct if u 1 ̸ = l ∗ u 2 for all l ∈ Z. Now, we are ready to state the main results. Theorem 1. Suppose that conditions (V 1 ), (f 1 )–(f 4 ) are satisfied. Then, one has the following conclusions. (1) If either σ = 1 and β ̸ =∞ or σ = −1 and α ̸ = −∞, then (7) has at least a nontrivial ground state solution. (2) If either σ = 1 and β = ∞ or σ = −1 and α = −∞, then (7) has no nontrivial solution. Theorem 2. Suppose that conditions (V 1 ), (f 1 )–(f 4 ) are satisfied and f n is odd in u. If either σ = 1 and β ̸ =∞ or σ = −1 andα ̸ = −∞, then (7) has infinitelymany pairs of geometrically distinct solutions. In what follows, we always assume that σ = 1. The other case can be reduced to σ = 1 by switching L to −L and ω to −ω. Remark 3. In [8], the author considered (7) with f n defined by f n (u) = |u| 2 u, (22) which obviously satisfies (f 1 )–(f 4 ); the author also discussed the case where f satisfies the Ambrosetti-Rabinowitz condition; that is, there exists μ > 2 such that 0 < μF n (u) ≤ f n (u) u, u ̸ = 0. (23) Clearly, (23) implies that F n (u) ≥ c|u| μ > 0 for |u| ≥ 1. So, it is a stronger condition than (f 3 ). Remark 4. In [9], the author assumed that f n satisfies the following condition: there exists θ ∈ (0, 1) such that 0 < u −1 f n (u) ≤ θf 󸀠 n (u) , u ̸ = 0. (24) Obviously, (24) implies (23) with μ = 1 + (1/θ), so it is a stronger condition than the Ambrosetti-Rabinowitz condition. In our paper, the nonlinearities satisfy more general superlinear assumptions instead of (24) which also implies (f 4 ). However, we do not assume that f n is differentiable and satisfies (24),M is not aC manifold ofE, and theminimizers onMmaynot be critical points of J. Hence, themethod of [9] does not apply anymore. Nevertheless,M is still a topological manifold, naturally homeomorphic to the unit sphere in E+ (see in detail in Section 3). We use the generalized Nehari manifold approach developed by Szulkin and Weth which is based on reducing the strongly indefinite variational problem to a definite one and prove that the minimizers of J onM are indeed critical points of J. Remark 5. In [7], it is shown that (7) has at least a nontrivial solution u ∈ l2 if f satisfies (V 1 ), (f 2 ), (f 3 ), and the following conditions: (B 1 ) F n (u) ≥ 0 for any u ∈ R andH n (u) := (1/2)f n (u)u − F n (u) > 0 if u ̸ = 0, (B 2 ) H n (u) → ∞ as |u| → ∞, and there exist r 0 > 0 and γ > 1 such that |f n (u)| γ /|u| γ ≤ c 0 H n (u) if |u| ≥ r 0 , where c 0 is a positive constant, In our paper, we use (9) and (f 4 ) instead of (B 1 ) and (B 2 ). 3. Proofs of Main Results We assume that (V 1 ) and (f 1 )–(f 4 ) are satisfied from now on. Lemma 6. F n (u) > 0 and (1/2)f n (u)u > F n (u) for all u ̸ = 0. Proof. By (f 2 ) and (f 4 ), it is easy to get that F n (u) > 0 ∀u ̸ = 0. (25) SetH n (u) = (1/2)f n (u)u − F n (u). It follows from (f 4 ) that H n (u) = u 2 f n (u) − ∫ u